We study a family of approximations to Euler's equation depending on two parameters
$\epsilon,η \ge 0$. When $\epsilon = η = 0$ we have Euler's equation and when both are positive we have
instances of the class of integro-differential equations called EPDiff in imaging science. These
are all geodesic equations on either the full diffeomorphism group
${Diff}_{H^\infty}(\mathbb{R}^n)$ or, if $\epsilon = 0$, its volume preserving subgroup. They are
defined by the right invariant metric induced by the norm on vector fields given by
$$ ||v||_{\epsilon,η} = \int_{\mathbb{R}^n} \langle L_{\epsilon,η} v, v \rangle\, dx $$
where $L_{\epsilon,η} = (I-\frac{η^2}{p} \triangle)^p \circ (I-\frac {1}{\epsilon^2} \nabla \circ div)$.
All geodesic equations are locally well-posed, and the $L_{\epsilon,η}$-equation admits solutions for
all time if $η > 0$ and $p\ge (n+3)/2$. We tie together solutions of all these equations by
estimates which, however, are only local in time. This approach leads to a new notion of momentum
which is transported by the flow and serves as a generalization of vorticity. We also discuss how
delta distribution momenta lead to ``vortex-solitons", also called ``landmarks" in imaging science,
and to new numeric approximations to fluids.

Peter W. Michor and David Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach,, Applied and Computational Harmonic Analysis, 23 (2007), 74.
doi: 10.1016/j.acha.2006.07.004.Google Scholar

Peter W. Michor and David Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach,, Applied and Computational Harmonic Analysis, 23 (2007), 74.
doi: 10.1016/j.acha.2006.07.004.Google Scholar